Theorems · Theorem · functional analysis
Submodule.closed_of_finiteDimensional
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : NontriviallyNormedField 𝕜] [CompleteSpace 𝕜] [inst_2 : AddCommGroup E]
[inst_3 : TopologicalSpace E] [IsTopologicalAddGroup E] [inst_5 : Module 𝕜 E] [ContinuousSMul 𝕜 E] [T2Space E]
(s : Submodule 𝕜 E) [FiniteDimensional 𝕜 ↥s], IsClosed ↑sA finite-dimensional subspace is closed.
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 173 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement and proof · cited by 24,529
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- SetLike.coestatement · cited by 8,199
- Submodulestatement and proof · cited by 7,192
- CompleteSpacestatement and proof · cited by 2,532
- FiniteDimensionalstatement and proof · cited by 1,854
- IsClosedstatement · cited by 1,639
- IsTopologicalAddGroupstatement and proof · cited by 1,394
- T2Spacestatement and proof · cited by 1,351
- ContinuousSMulstatement and proof · cited by 1,016
Cited by8
Results whose statement or proof uses this declaration.
- MeasureTheory.Measure.addHaar_submoduleproof · cited by 3
- FiniteDimensional.of_totallyBounded_nhds_zeroproof · cited by 3
- LinearMap.isClosedEmbedding_of_injectiveproof · cited by 3
- AffineSubspace.closed_of_finiteDimensionalproof · cited by 2
- Submodule.topologicalClosure_eq_selfproof · cited by 2
- Submodule.isClosed_mono_of_finiteDimensional_quotientproof · cited by 1
- Submodule.isClosed_sup_finiteDimensionalproof · cited by 1
- exists_norm_le_le_norm_sub_of_finsetproof · cited by 1